ON THE MOTION OF PENDULUMS. 141
wrong law in the reduction, may be considered to have been accounted for.
The very accurate experiments of Bessel are unfortunately not available for more than a very rude comparison, for the reason mentioned in the foot-note at p. 97.
As regards the correction on account of the air to the time of vibration of a pendulum, we have seen that in the case of a sphere, and very approximately in the case of a long cylindrical rod which is not extremely narrow, it is of the form
where G and H depend on the form of the pendulum, but not upon the pressure, nor indeed on the nature, of the. surrounding gas, which might be other than air. There can be little doubt that the same would apply as a very near approximation to any of the ordinary forms of pendulum, though in that case the constants G, H cannot in general be obtained by calculation. The first term depends partly on buoyancy, partly on the inertia of the gas regarded as a perfect fluid. As the latter part cannot be calculated, there is no need to calculate the former, since the two have to be determined as a whole by observation. As the value of p for air is now well known, the constants G and // may be determined from the differences in the times of vibration at three suitably distributed pressures. These constants are determined once for all for the same pendulum. They may even be applied without a fresh experimental determination to any other pendulum of which the external form is geometrically similar, even though the internal distribution of mass be different, of course with due regard to the dimensions of the terms with respect to the units of length and time. Moreover unless we want to combine observations with different gases, or else to take account of the variation of p. with temperature, we may write the above formula
and as we must appeal to experiment for the determination of H', we do not even need to know the value of ^.